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- Thread starter nycmathguy
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Mark44

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Plot some points. What are f(0), f(1/4), f(1/2), f(2), f(2.5), etc.?Homework Statement::Investigate each limit.

Relevant Equations::See attachment for function.

Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.

Also, you've posted a few threads just now with little or no work shown. That's a violation of forum rules. You have to show

- #3

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1) What is f(n)

2) What is f(n+1)

3) What is f(x) for every ##x\in(n,n+1)## for example for x=(2n+1)/2 the midpoint of n and n+1.

- #4

Sorry but I don't get it. Still lost.

1) What is f(n)

2) What is f(n+1)

3) What is f(x) for every ##x\in(n,n+1)## for example for x=(2n+1)/2 the midpoint of n and n+1.

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what is f(1) and f(2) equal to for example? Hint: 1 and 2 are integers

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I say for (1), the answer is 0.

The answer for (2) is 1.

Yes?

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Νο, ##f(n)=f(n+1)=1## for all integers n. The function definition tells us that f(x)=1 if x is integer.I say for (1), the answer is 0.

The answer for (2) is 1.

Yes

- #9

For (1), x tends to an integer. Thus, then f(x) = 1.Νο, ##f(n)=f(n+1)=1## for all integers n. The function definition tells us that f(x)=1 if x is integer.

For (2), x tends to a rational number. Thus, f(x) = 0.

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What about the following two cases using the same attachment?Νο, ##f(n)=f(n+1)=1## for all integers n. The function definition tells us that f(x)=1 if x is integer.

Investigate each limit.

1. lim f(x) x→3

2. lim f(x) x→0

For (1), x tends to an integer. Thus, f(x) = 1.

For (2), x tends to 0, which is not an integer.

Thus, f(x) = 1.

Yes?

- #12

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f(1)=f(0)=1

f(x)=0 for all x inbetween 0 and 1.

Then what do you think is the ##\lim_{x\to 0} f(x)## (or ##\lim_{x\to 1} f(x)##..

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For both cases the limit is 0. (0 is an integer btw).What about the following two cases using the same attachment?

Investigate each limit.

1. lim f(x) x→3

2. lim f(x) x→0

For (1), x tends to an integer. Thus, f(x) = 1.

For (2), x tends to 0, which is not an integer.

Thus, f(x) = 1.

Yes?

- #14

Homework Statement::Investigate each limit.

Relevant Equations::See attachment for function.

Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.

Can you elaborate a little more?

f(1)=f(0)=1

f(x)=0 for all x inbetween 0 and 1.

Then what do you think is the ##\lim_{x\to 0} f(x)## (or ##\lim_{x\to 1} f(x)##..

It's just not sinking in. In fact, Sullivan stated in his book that this is considered a challenging problem.

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hm ok let me see

If I tell you that f(x)=0 for all x then what is the ##\lim_{x\to 0} f(x)##.

If I tell you that f(x)=0 for all x then what is the ##\lim_{x\to 0} f(x)##.

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- #17

In that case, it is 0.hm ok let me see

If I tell you that f(x)=0 for all x then what is the ##\lim_{x\to 0} f(x)##.

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Correct now let's say I tweak the function and the function f is now f(x)=0 for all x EXCEPT for x=0 which I define to be f(0)=1. Do you think that the above limit changes or remains the same?In that case, it is 0.

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You said except for x = 0. I say the limit is 1?Correct now let's say I tweak the function and the function f is now f(x)=0 for all x EXCEPT for x=0 which I define to be f(0)=1. Do you think that the above limit changes or remains the same?

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Nope it isn't 1. What is f(x) equal to ,as x tends to 0, for example what is f(0.5), f(0.4), f(0.3) , f(0.2) and so on..You said except for x = 0. I say the limit is 1?

- #21

So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.Nope it isn't 1. What is f(x) equal to ,as x tends to 0, for example what is f(0.5), f(0.4), f(0.3) , f(0.2) and so on..

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That's correct. So what conclusion can you make from this? where does f(x) tend to as x tends to 0?So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.

- #23

So, f(x) tends to 0 as x-->0.That's correct. So what conclusion can you make from this? where does f(x) tend to as x tends to 0?

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yes and this is true regardless of what value we choose to give to f(0). As long as f(x)=0 for all ##x\neq 0## .So, f(x) tends to 0 as x-->0.

- #25

Trust me, I plan to journey through calculus l,ll, and lll. We will see limit questions up the wall.yes and this is true regardless of what value we choose to give to f(0). As long as f(x)=0 for all ##x\neq 0## .

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- #27

This one is tricky.

I say the limit is 5.

- #28

Mark44

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No. f(x) = 1 if x is an integer, but for all other numbers, f(x) = 0.For (1), x tends to an integer. Thus, then f(x) = 1.

The question is asking about ##\lim_{x \to 2} f(x)##, not f(x). Even though f(2) = 1, ##\lim_{x \to 2} f(x)## is some other value.

Again, no.For (1), x tends to an integer. Thus, f(x) = 1.

First off, 0For (2), x tends to 0, which is not an integer.

Thus, f(x) = 1.

Most "decimal" numbers are not rational (e.g., ##\pi \approx 3.141592## and ##\sqrt 2 \approx 1.414##), and some rational numbersSo, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.

Just to check your understanding, if i tell you f(x)=5 for all x≠0 and f(0)=10, what is the limit of f(x) as x tends to 0?

Right, but it's not tricky if you understand the idea of what a limit means.This one is tricky.

I say the limit is 5.

- #29

Ok. There are many more limits coming our way in time. This is just the beginning of the long journey.No. f(x) = 1 if x is an integer, but for all other numbers, f(x) = 0.

The question is asking about ##\lim_{x \to 2} f(x)##, not f(x). Even though f(2) = 1, ##\lim_{x \to 2} f(x)## is some other value.

Again, no.

First off, 0isan integer. Second, you're again not distinguishing between function values (e.g. f(0)) and values of the limit. Here the limit expression is ##\lim_{x \to 1/2} f(x)##, which just happens to be the same as f(1/2).

Most "decimal" numbers are not rational (e.g., ##\pi \approx 3.141592## and ##\sqrt 2 \approx 1.414##), and some rational numbersareintegers (e.g., 2/1, 6/2, and so on).

Right, but it's not tricky if you understand the idea of what a limit means.

- #30

Mark44

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So make sure you understand the difference between, say, ##f(c)## and ##\lim_{x \to c} f(x)##. For a continuous function f, they will be the same, but not necessarily so for discontinuous or piecewise-defined functions.Ok. There are many more limits coming our way in time.

- #31

Will do.So make sure you understand the difference between, say, ##f(c)## and ##\lim_{x \to c} f(x)##. For a continuous function f, they will be the same, but not necessarily so for discontinuous or piecewise-defined functions.

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